Synergistic control of oscillations in the NF-kappaB signalling pathway

In previous work, we studied the behaviour of a model of part of the NF-kappaB signalling pathway. The model displayed oscillations that varied both in number, amplitude and frequency when its parameters were varied. Sensitivity analysis showed that just nine of the 64 reaction parameters were mainly responsible for the control of the oscillations when these parameters were varied individually. However, the control of the properties of any complex system is distributed, and, as many of these reactions are highly non-linear, we expect that their interactions will be too. Pairwise modulation of these nine parameters gives a search space some 50 times smaller (81 against 4096) than that required for the pairwise modulation of all 64 reactions, and this permitted their study (which would otherwise have been effectively intractable). Strikingly synergistic effects were observed, in which the effect of one of the parameters was strongly (and even qualitatively) dependent on the values of another parameter. Regions of parameter space could be found in which the amplitude, but not the frequency (timing), of oscillations varied, and vice versa. Such modelling will permit the design and performance of experiments aimed at disentangling the role of the dynamics of oscillations, rather than simply their amplitude, in determining cell fate. Overall, the analyses reveal a level of complexity in these dynamic models that is not apparent from study of their individual parameters alone and point to the value of manipulating multiple elements of complex networks to achieve desired physiological effects.     

Model of calcium oscillations due to negative feedback in olfactory cilia

We present a mathematical model for calcium oscillations in the cilia of olfactory sensory neurons. The underlying mechanism is based on direct negative regulation of cyclic nucleotide-gated channels by calcium/calmodulin and does not require any autocatalysis such as calcium-induced calcium release. The model is in quantitative agreement with available experimental data, both with respect to oscillations and to fast adaptation. We give predictions for the ranges of parameters in which oscillations should be observable. Relevance of the model to calcium oscillations in other systems is discussed.     

Allosteric oscillatory enzymes: influence of the number of protomers on metabolic periodicities.

The study of a concerted allosteric model for an enzyme activated by the reaction product shows that this system can generate sustained metabolic oscillations regardless of the number of protomers constituting the enzyme. The analysis extends the results previously obtained in a dimeric model for the phosphofructokinase reaction which produces glycolytic periodicities. When the substrate and product concentrations evolve on comparable time scales, the amplitude of oscillations significantly drops as the number of enzyme subunits evolves from 2 to 8. The width of the domain of substrate injection rates which produce oscillations and the periodic variation in enzyme activity also depend on the number of protomers and on the time scale structure of the system. Theoretical predictions are compared with the experiments on glycolytic oscillations in yeast and muscle, and with the structural characteristics of phosphofructokinase. The results are also discussed in relation with the mechanism of cyclic AMP oscillations in the slime mold Dictyostelium discoideum.     

Oscillations in peroxidase-catalyzed reactions and their potential function in vivo

The peroxidase-oxidase reaction has become a model system for the study of oscillations and complex dynamics in biochemical systems. In the present paper we give an overview of previous experimental and theoretical studies of the peroxidase-oxidase reaction. Recent in vitro experiments have raised the question whether the reaction also exhibits oscillations and complex dynamics in vivo. To investigate this possibility further we have undertaken new experimental studies of the reaction, using horseradish extracts and phenols which are widely distributed in plants. The results are discussed in light of the occurrence and a possible functional role of oscillations and complex dynamics of the peroxidase-oxidase reaction in vivo.     

Mechanical and biochemical modeling of cortical oscillations in spreading cells

Actomyosin-based cortical contractility is a common feature of eukaryotic cells and is involved in cell motility, cell division, and apoptosis. In nonmuscle cells, oscillations in contractility are induced by microtubule depolymerization during cell spreading. We developed an ordinary differential equation model to describe this behavior. The computational model includes 36 parameters. The values for all but two of the model parameters were taken from experimental measurements found in the literature. Using these values, we demonstrate that the model generates oscillatory behavior consistent with current experimental observations. The rhythmic behavior occurs because of the antagonistic effects of calcium-induced contractility and stretch-activated calcium channels. The model makes several experimentally testable predictions: 1), buffering intracellular calcium increases the period and decreases the amplitude of cortical oscillations; 2), increasing the number or activity of stretch activated channels leads to an increase in period and amplitude of cortical oscillations; 3), inhibiting Ca²⁺ pump activity increases the period and amplitude of oscillations; and 4), a threshold exists for the calcium concentration below which oscillations cease.     

Allosteric oscillatory enzymes : Influence of the number of protomers on metabolic periodicities

The study of a concerted allosteric model for an enzyme activated by the reaction product shows that this system can generate sustained metabolic oscillations regardless of the number of protomers constituting the enzyme. The analysis extends the results previously obtained in a dimeric model for the phosphofructokinase reaction which produces glycolytic periodicities. When the substrate and product concentrations evolve on comparable time scales, the amplitude of oscillations significantly drops as the number of enzyme subunits evolves from 2 to 8. The width of the domain of substrate injection rates which produce oscillations and the periodic variation in enzyme activity also depend on the number of protomers and on the time scale structure of the system. Theoretical predictions are compared with the experiments on glycolytic oscillations in yeast and muscle, and with the structural characteristics of phosphofructokinase. The results are also discussed in relation with the mechanism of cyclic AMP oscillations in the slime mold Dictyostelium discoideum.     

A temperature-compensated model for circadian rhythms that can be entrained by temperature cycles

From the viewpoint that reaction rates will change with temperature, we present a general method to build circadian clock models that generate circadian oscillations with almost constant period under different constant ambient temperature, and propose an algorithm estimating the parameter condition for compensated period against the change of temperature based on the PER single-feedback loop model of Goldbeter [1995. A model for circadian oscillations in the Drosophila period protein (PER). Proc. R. Soc. London Ser. B 261, 319-324] for Drosophila. We show that the model with derived parameters can realize the temperature compensation over a wide range of temperature, and simultaneously can realize the entrainment to temperature cycles.     

A model of stochastic variations of postsynaptic activity

Stochastic oscillations imitating postsynaptic activity in the excitatory neurons are produced by a nonlinear difference equation which does not contain any sources of noise. The given back inhibition via inhibitory interneurons presents a negative feedback loop due to which oscillations in the model system are realized. By means of variation of parameters of the system the patterns of stochastic oscillations can be changed in wide range of physiologically meaningful patterns of the neuronal activity.     

Oscillatory kinetics of the peroxidase-oxidase reaction in an open system. Experimental and theoretical studies.

1. The oscillations in the peroxidase (donor: hydrogen-peroxide oxidoreductase, EC 1.11.1.7)-catalyzed reaction between NADH and O2 are undamped when the reaction is carried out in a system open to both substrates and when 2,4-dichlorophenol and methylene blue are present in the solution. 2. The waveform of the oscillations changes when the concentration of peroxidase is varied. 3. The waveforms obtained experimentally can be simulated by a branched chain reaction model in which the branching is quadratic. 4. A correlation between the present knowledge of the reaction and the model can be made by combining well established and hypothetical reaction steps into a few reaction schemes. A selection among schemes however, is not possible at the present time. 5. Compound III participates in the reaction as an active intermediate. This is possible because dichlorophenol stimulates the break down of compound III.     

Diffusion induced oscillatory insulin secretion

Oscillatory secretion of insulin has been observed in many different experimental preparations. Here we examine a mathematical model for in vitro insulin secretion from pancreatic beta cells in a flow-through reactor. The analysis shows that oscillations result because of an important interplay between flow rate of the reactor and insulin diffusion. In particular, if the ratio of flow rate to volume of the reaction bed is too large, oscillations are eliminated, in contradiction to the conclusions of Maki and Keizer (L. W. Maki and Keizer J. Mathematical analysis of a proposed mechanism for oscillatory insulin secretion in perifused HIT-15 cells. Bull. Math. Biol., 57 (1995), 569–591). Furthermore, with reasonable numbers for the experimental parameters and the diffusion of insulin, the model equations do not exhibit oscillations.     

Oscillatory enzyme reactions and Michaelis–Menten kinetics

Oscillations occur in a number of enzymatic systems as a result of feedback regulation. How Michaelis–Menten kinetics influences oscillatory behavior in enzyme systems is investigated in models for oscillations in the activity of phosphofructokinase (PFK) in glycolysis and of cyclin-dependent kinases in the cell cycle. The model for the PFK reaction is based on a product-activated allosteric enzyme reaction coupled to enzymatic degradation of the reaction product. The Michaelian nature of the product decay term markedly influences the period, amplitude and waveform of the oscillations. Likewise, a model for oscillations of Cdc2 kinase in embryonic cell cycles based on Michaelis–Menten phosphorylation–dephosphorylation kinetics shows that the occurrence and amplitude of the oscillations strongly depend on the ultrasensitivity of the enzymatic cascade that controls the activity of the cyclin-dependent kinase.     

Computer simulation of damped oscillations during peroxidase-catalyzed oxidation of indole-3-acetic acid

Oscillation patterns in horseradish peroxidase (HRP)-catalyzed oxidation of indole-3-acetic acid (IAA) at neutral pH were studied using computer simulation. Under certain conditions, such as the presence of a reaction promoter and continuous intake of oxygen from the gaseous phase, the simulated system exhibits damped oscillations of the concentrations of oxygen in the aqueous phase, [O(2)](aq), and of all the reaction intermediates. The critical concentration of oxygen in aqueous phase, [O(2)](cr)(aq), was used to describe the nature of the oscillations. The critical concentration is the concentration at which the system abruptly changes its properties. If [O(2)](aq) is higher than [O(2)](cr)(aq) then the reaction develops as an avalanche, otherwise, the reaction stops. The nature of oscillations is accounted for by the interaction of two processes: the consumption/accumulation of oxygen and the accumulation/consumption of reaction intermediates. Oscillations are always damped. Neither HRP or umbelliferone (Umb) deactivation nor IAA consumption can account for the damping. The nature of the damping is determined by the termination reactions of free radical intermediates and ROOH. The three major parameters of oscillations: period of oscillations, initial amplitude of oscillations and the rate of damping were studied as functions of: (i) oxygen concentration in the gaseous phase, (ii) initial oxygen concentration in aqueous phase, (iii) the concentration of IAA and (iv) the initial concentration of HRP.     

Quasi-periodic behaviour in a model for the lithium-induced,electrical oscillations of frog skin

The fact that oscillations can be induced in studies of the maintenance of the electrical potential of frog skin by addition of lithium allowed evaluation of several parameters fundamental to the functioning of the system in vivo (e.g. relative volumes of internal compartments, characteristic times of ionic exchanges between compartments). A realistic model was thus proposed under the form of a set of ordinary differential equations. In the past, numerical simulations using such a model reproduced the periodic experimental oscillations and was able to provide an explanation for the global synchronised oscillations of the whole skin. In that paper, new numerical simulations reproduce the non-periodic oscillations which were observed two decades ago, but not reproduced by the model. Moreover, the dynamical process under which all the local oscillators are synchronised is explained in terms of a tangent bifurcation.     

On spontaneous discharge obtained from a physicochemical model of excitation

It is shown that a slight modification of a model of excitatory phenomena in irritable tissues, which has been treated before, exhibits spontaneous oscillations. The frequency of these oscillations and the time-course of the potential across the model membrane have been determined, together with the dependence of some of their characteristics on some important parameters, particularly (Ca⁺⁺).     

A study of the regulation of photosynthesis by the analysis of the effect of kinetic parameters on the characteristics of the oscillatory regime and the rate of CO2 assimilation

Earlier at the biophysics department, the experimental data on the oscillations of delayed luminescence have been described with the help of a mathematical model. Here we studied the influence of the model parameters on the characteristics of the oscillatory regime. The frequencies and damping factors of the oscillations at different parameter values were calculated using the Lyapunov analysis. It was shown that, in addition to oscillations observed experimentally, other, rapidly damping oscillations may exist. The dependence of the CO2 assimilation rate on the model parameters was studied. It was shown that the intensity of the light absorbed by photosystems I and II may differently affect the assimilation of CO2.     

Mathematical models of endocrine systems

There is proposed a generalized mathematical model of endocrine systems, consisting of a set of differential equations which describe a chain of chemical reactions. The product of each reaction stimulates or inhibits some other reaction in the chain except possibly the last, which may or may not influence the system. At least one reaction must be independent and able to proceed without stimulation or inhibition by the products of other reactions. If only two reactions of the type assumed constitute a closed chain, sustained periodic variations in the concentrations of the reaction products cannot occur. If the chain consists of three or more reactions forming a closed loop, sustained oscillations, such as are observed in the menstrual cycle or in the mental disorder called periodic catatonia, can occur under suitable conditions. In this case, the concentrations of the system components exhibit relaxation oscillations characterized by periodic degeneration of the system when an independent reaction becomes completely inhibited by other reaction products. A set of conditions sufficient to produce periodicities in component concentrations is presented. Application of the model to the normally periodic system of the menstrual cycle and to the abnormal endocrine system which causes periodic catatonia is discussed.     

From bistability to oscillations in a model for the isocitrate dehydrogenase reaction

Considered is a bienzymatic system consisting of isocitrate dehydrogenase (IDH, EC 1.1.1.42), which transforms NADP(+) into NADPH, and of diaphorase (DIA, EC 1.8.1.4), which catalyzes the reverse reaction. Experimental evidence as well as a theoretical model show the possibility of a coexistence between two stable steady states in this reaction system. The phenomenon originates from the regulatory properties of IDH. We extend the analysis of a theoretical model proposed for the IDH-DIA bienzymatic system and investigate the occurrence of different modes of bistability, with or without hysteresis, i.e. in the presence of two or only one limit point bounding the domain of multiple steady states. The analysis indicates that the two types of bistability may sometimes be observed sequentially as a given control parameter is progressively increased. We further obtain conditions in which sustained oscillations develop in the model. These results establish the isocitrate dehydrogenase reaction coupled to diaphorase as a suitable candidate for further experimental and theoretical studies of bistability and oscillations in biochemical systems.     

On the role of enzyme cooperativity in metabolic oscillations: analysis of the Hill coefficient in a model for glycolytic periodicities.

The role of enzyme cooperativity in the mechanism of metabolic oscillations is analyzed in a concerted allosteric model for the phosphofructokinase reaction. This model of a dimer enzyme activated by the reaction product accounts quantitatively for glycolytic periodicities observed in yeast and muscle. The Hill coefficient characteristic of enzyme-substrate interactions is determined in the model, both at the steady state and in the course of sustained oscillations. Positive cooperativity is a prerequisite for periodic behavior. A necessary condition for oscillation in a dimer K system is a Hill coefficient larger than 1.6 at the unstable stationary state. The analysis suggests that positive as well as negative effectors of phosphofructokinase inhibit glycolytic oscillations by inducing a decrease in enzyme cooperativity. The results are discussed with respect to glycolytic and other metabolic periodicities.     

Modeling the mammalian circadian clock: sensitivity analysis and multiplicity of oscillatory mechanisms

We extend the study of a computational model recently proposed for the mammalian circadian clock (Proc. Natl Acad. Sci. USA 100 (2003) 7051). The model, based on the intertwined positive and negative regulatory loops involving the Per, Cry, Bmal1, and Clock genes, can give rise to sustained circadian oscillations in conditions of continuous darkness. These limit cycle oscillations correspond to circadian rhythms autonomously generated by suprachiasmatic nuclei and by some peripheral tissues. By using different sets of parameter values producing circadian oscillations, we compare the effect of the various parameters and show that both the occurrence and the period of the oscillations are generally most sensitive to parameters related to synthesis or degradation of Bmal1 mRNA and BMAL1 protein. The mechanism of circadian oscillations relies on the formation of an inactive complex between PER and CRY and the activators CLOCK and BMAL1 that enhance Per and Cry expression. Bifurcation diagrams and computer simulations nevertheless indicate the possible existence of a second source of oscillatory behavior. Thus, sustained oscillations might arise from the sole negative autoregulation of Bmal1 expression. This second oscillatory mechanism may not be functional in physiological conditions, and its period need not necessarily be circadian. When incorporating the light-induced expression of the Per gene, the model accounts for entrainment of the oscillations by light-dark (LD) cycles. Long-term suppression of circadian oscillations by a single light pulse can occur in the model when a stable steady state coexists with a stable limit cycle. The phase of the oscillations upon entrainment in LD critically depends on the parameters that govern the level of CRY protein. Small changes in the parameters governing CRY levels can shift the peak in Per mRNA from the L to the D phase, or can prevent entrainment. The results are discussed in relation to physiological disorders of the sleep-wake cycle linked to perturbations of the human circadian clock, such as the familial advanced sleep phase syndrome or the non-24h sleep-wake syndrome.